Lecture P2: Integral Momentum Equation

Wow! The 16.070 problem set must have taken its toll. When I started the lecture there were 23 people in the room. When I finished there were only 42. This will cause some people to struggle with the application of the integral momentum theorem. This is one of the most important concepts in the propulsion material. If you are one of these people: 1) read the notes, 2) start your homework early so you can ask me questions, and 3) come to the recitation on Tuesday--I will do some examples.

I think the discussion of the integral momentum equation went pretty well. I was pleased that the class identified problems in each of the 3 PRS questions (see my notes on these PRS #1, PRS#2, PRS#3). It is gratifying when you folks know enough to not only answer the problem, but also to find what is wrong with the problem statement.

Responses to 'Muddiest Part of the Lecture Cards'

(21 respondents)

1) How did you introduce (pe-po)Ae into the thrust equation? (3 students) Read over the section of the notes and the explanation of the PRS question. These should help. Pressure forces are one of the external forces (along with drag, thrust, gravity, etc.) that we typically work with. They are captured in the SFx term in the integral momentum equation. Since they are applied on a surface (e.g. the surface of the control volume), you need to do the surface integral around the control volume and add up the net force. Remember that pressure always acts normal to a surface (so if you know the orientation of the surface, you know the orientation of the force.

2) In the second concept question, if inlet and exhaust velocities are the same regardless of back pressure, how can you manipulate back pressure? (1 student). They are not the same regardless of back pressure, but I wanted the class to focus on arriving at the pressure term in the thrust equation through simplified pressure balance arguments. As I noted, I was so intent on this objective, that I didn't even include in the problem statement the proviso that everything else was constant. If you decrease Pe, won't this increase mass flow because more mass is crammed into the engine? (1 student) Changing the back pressure can change the operation of the engine but I was trying to highlight the pressure force term. the actual influence of back pressure on an engine can be quite complicated.

3) For that PRS question with the plane taking off, the answer makes sense, but I thought when we were talking about the streamtubes that you said inlet velocity was constant. That would lead to constant thrust. (1 student) Good question. The velocity in the face of the inlet is roughly constant (for constant fuel flow into the engine and thus constant mass flow into the engine). However, as the aircraft changes speed the area of the streamtube captured by the inlet changes (to maintain mass flow as the flight velocity changes). But the velocity we work with in the thurst equation is not the velocity on the face of the inlet, it is the flight velocity (a much more convenient parameter directly related to vehicle performance).

4) I am confused about where the terms come from (mduo/dt). (Notations written on card--yes?). (1 student) The question refers to the example of the falling eraser. You have it exactly correct as written on your card.

5) In the equation SFx - Fox = ...., does the Fx mean just the forces in the x-direction? (1 student). Yes. This equation is one of three (x, y, z components) from the integral momentum equation. The momentum equation is a vector relation and expresses force and momentum balances in three coordinate directions. To get the version for the y-component, just replace all the little "x's" with "y's".

6) Can we stop using screwed up coordinate systems (drawing of axes with x-direction pointed up) and just stick with the usual ones? (1 student). We don't design bridges in this department. We design things that fly and spin. Coordinate frames can appear in all sorts of orientations.

7) [On the 3rd PRS question] Since the boundary line is parallel to the y-axis why is the flux in the y-direction not zero? (1 student) Good question. The problem asks for the flux of y-momentum across the interface. The fluid crossing the interface (in the x-direction) carries with it some y-momentum (i.e. it is moving up and over at the same time).

8) Can we see some more examples of applying the integral momentum equation? (1 student) Yes!

9)Why is an engine most efficient when the back pressure is equal to the atmospheric pressure? (1 student). This is a super question. The thrust force produced by the change in momentum flux of the gas flowing through the control volume is transmitted to the engine surfaces through pressure and viscous forces. It is easier for us to calculate thrust by looking at the change in momentum flux + pressure on all of the surfaces of the gas that goes through the engine, but we could just as well integrate up all the surface forces on every internal part of the engine and sum all these up and get the same answer. However, given the complexity of the geometry and flowfield within an engine, this is not practical. But nonetheless, when the nozzle exit velocity and/or pressure changes, there is a corresponding change in pressure on the surfaces within the engine and this in turn leads to a change in thrust. If the flow is subsonic at the exit of the nozzle, then the pressures are matched (the ambient pressure and the exhaust pressure equilibrate--just like opening a high pressure air bottle in a room). It is only for the case of supersonic nozzle flow that this term can be important. For this situation, the typical case is one where the flow has not fully expanded to atmospheric pressure (pe>po). The flow does expand after leaving the engine, but there is no structure for it to push against, so the expansion produces no thrust. There is a pretty good discussion of this on page 79 of Kerrebrock's book "Aircraft Engines and Gas Turbines" (available in the library--it is the text for 16.50). However there is no need to worry about this--it is a more advanced topic.

10) No mud (9 students). Good!